OFFSET
0,1
COMMENTS
The denominators of the T(n, n+1) with T(0, m) = A164555(m)/A027642(m) and T(n, m) = T(n-1, m+1) - T(n-1, m), n >= 1, m >= 0. For the numerators of the T(n, n+1) see A191972.
A164555(n)/A027642(n) is an autosequence (eigensequence whose inverse binomial transform is the sequence signed) of the second kind; the main diagonal T(n, n) is twice the first upper diagonal T(n, n+1).
Also the denominators of T(n, n+1) of the table defined by A085737(n)/A085738(n), the upper diagonal, called the median Bernoulli numbers by Chen. As such, Chen proved that a(n) is even only for n=0 and n=1 and that a(n) are squarefree numbers. (see Chen link). - Michel Marcus, Feb 01 2013
The sum of the antidiagonals of T(n,m) is 1 in the first antidiagonal, otherwise 0. Paul Curtz, Feb 03 2015
REFERENCES
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..450
Kwang-Wu Chen, A summation on Bernoulli numbers, Journal of Number Theory, Volume 111, Issue 2, April 2005, Pages 372-391.
Peter Luschny, Computation and asymptotics of the Bernoulli numbers
FORMULA
T(n, n) = 2*T(n, n+1).
EXAMPLE
The first few rows of the T(n, m) array (difference table of the Bernoulli numbers) are:
1, 1/2, 1/6, 0, -1/30, 0, 1/42,
-1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42,
1/6, 1/6, 2/15, 1/15, -1/105, -1/21, -1/105,
0, -1/30, -1/15, -8/105, -4/105, 4/105, 8/105,
-1/30, -1/30, -1/105, 4/105, 8/105, 4/105, -116/1155,
0, 1/42, 1/21, 4/105, -4/105, -32/231, -16/231,
1/42, 1/42, -1/105, -8/105, -116/1155, 16/231, 6112/15015,
MAPLE
T := proc(n, m)
option remember;
if n < 0 or m < 0 then
0 ;
elif n = 0 then
if m = 1 then
-bernoulli(m) ;
else
bernoulli(m) ;
end if;
else
procname(n-1, m+1)-procname(n-1, m) ;
end if;
end proc:
A190339 := proc(n)
denom( T(n+1, n)) ;
end proc: # R. J. Mathar, Apr 25 2013
MATHEMATICA
nmax = 23; b[n_] := BernoulliB[n]; b[1]=1/2; bb = Table[b[n], {n, 0, 2*nmax-1}]; diff = Table[Differences[bb, n], {n, 1, nmax}]; Diagonal[diff] // Denominator (* Jean-François Alcover, Aug 08 2012 *)
PROG
(Sage)
def A190339_list(n) :
T = matrix(QQ, 2*n+1)
for m in (0..2*n) :
T[0, m] = bernoulli_polynomial(1, m)
for k in range(m-1, -1, -1) :
T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
for m in (0..n-1) : print([T[m, k] for k in (0..n-1)])
return [denominator(T[k, k+1]) for k in (0..n-1)]
A190339_list(7) # Also prints the table as displayed in EXAMPLE. Peter Luschny, Jun 21 2012
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Paul Curtz, May 09 2011
EXTENSIONS
Edited and Maple program added by Johannes W. Meijer, Jun 29 2011, Jun 30 2011
New name from Peter Luschny, Jun 21 2012
STATUS
approved